Andy - congratulations on getting a 1st from Durham University. Most consulting firms (the vast majority) will view this as putting you high enough up the academic standings to secure an interview. A few of the most elite firms will be looking for people with a 1st from Oxbridge. This is really a reflection of the fact that they are only recruiting a dozen or so graduates each year and so can afford to be this elitist (and also because they only have such a modest intake, they don't need to spread themselves beyond Oxbridge to get the hires they need).
I would suggest all the big full-service firms would be interested in interviewing you - Accenture, PA Consulting, Capgemini, Deloitte, PwC, IBM, etc. The top-tier of strategy firms (McKinsey, Bain, BCG, Booz Allen, Roland Berger, Monitor, OC&C) tend to have the highest recruiting hurdles and with these firms you will find that some will interview you and others will not. Pretty much all the smaller firms will be interested in interviewing someone with a 1st from Durham.
My advice would be to interview with (and therefore apply to) a range of large firms and small firms. Once you go through the assessment days and interview procedures you will I'm sure be surprised that clear "winners" emerge. Some firms you will really take a liking to, because you are a great fit for their culture. Others that you might have been attracted to on paper you will suddenly feel are not nearly as interesting to you.
Hope that helps Andy - and good luck.